On a fundamental property of the cross- Gramian matrix

The cross-Gramian matrixW_{c0} (T)for linear single-input single-output (SISO) dynamical systems has been related to the controllability and observability Gramians viaW^{2}_{c0}(T)= W_{c}(T)W_{0}(T)forT= \infty. The result is now proved for any arbitrary time intervalT, and is extended to symmetric multivariable systems which have symmetric transfer function matrices.